Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imaiun1 Structured version   Visualization version   GIF version

Theorem imaiun1 43664
Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
Assertion
Ref Expression
imaiun1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem imaiun1
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3288 . . . 4 (∃𝑥𝐴𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑧𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
2 vex 3484 . . . . . 6 𝑦 ∈ V
32elima3 6085 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
43rexbii 3094 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
5 eliun 4995 . . . . . . 7 (⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵)
65anbi2i 623 . . . . . 6 ((𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ (𝑧𝐶 ∧ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵))
7 r19.42v 3191 . . . . . 6 (∃𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑧𝐶 ∧ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵))
86, 7bitr4i 278 . . . . 5 ((𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
98exbii 1848 . . . 4 (∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑧𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
101, 4, 93bitr4ri 304 . . 3 (∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
112elima3 6085 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵))
12 eliun 4995 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
1310, 11, 123bitr4i 303 . 2 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ 𝑦 𝑥𝐴 (𝐵𝐶))
1413eqriv 2734 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  cop 4632   ciun 4991  cima 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-iun 4993  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  trclimalb2  43739
  Copyright terms: Public domain W3C validator